No. In short, Torah DOES NOT allow people to convert, except the two specific nations. However, Torah's does provide ways for outsiders to join the faith. Let's take a look:
YLT Ex 12:48 'And when a sojourner sojourneth with thee, and hath made a passover to Jehovah, every male of his is to be circumcised, and then he doth come near to keep it, and he hath been as a native of the land, but any uncircumcised one doth not eat of it;
YLT Ex 12:49 one law is to a native, and to a sojourner who is sojourning in your midst.'
YLT Lv 19:34 as a native among you is the sojourner to you who is sojourning with you, and thou hast had love to him as to thyself, for sojourners ye have been in the land of Egypt; I am Jehovah your God.
YLT Num 9:14 'And when a sojourner sojourneth with you, then he hath prepared a passover to Jehovah, according to the statute of the passover, and according to its ordinance, so he doth; one statute is to you, even to a sojourner, and to a native of the land.'
YLT Num 15:13 every native doth thus with these, at bringing near a fire-offering of sweet fragrance to Jehovah;
YLT Num 15:14 and when a sojourner sojourneth with you, or whoso is in your midst to your generations, and he hath made a fire-offering of sweet fragrance to Jehovah, as ye do so he doth.
YLT Num 15:15 'One statute is for you of the congregation and for the sojourner who is sojourning, a statute age-during to your generations: as ye are so is the sojourner before Jehovah;
YLT Num 15:16 one law and one ordinance is to you and to the sojourner who is sojourning with you.'
So, as the verses above state, Torah allows ANYONE to sojourn in the midst of the Children of Israel. The only requirement is that the sojourner lives according to the Written Torah. For example, the best verse from ones above is Leviticus 19:34, where it says explicitly that "as a native among you is the sojourner to you who is sojourning with you", which means that the only difference between the sojourner and native is that the natives are entitled to the Land allotment, while a sojourner has to purchase what is available and they can't own their land. Otherwise, as Torah states, there is no difference.
Aside from that there are two more classes of people the Torah talks about. Let's take a look:
YLT Dt 23:7 'Thou dost not abominate an Edomite, for thy brother he is; thou dost not abominate an Egyptian, for a sojourner thou hast been in his land;
YLT Dt 23:8 sons who are begotten of them, a third generation of them, doth enter into the assembly of Jehovah.
So as you can see, only two nations were explicitly allowed to convert, namely Edomites and Egyptians. The first two generations of them would be considered as sojourners (gers). The third generation of them, however, could become Hebrews and be considered natives. They would be allowed to be a part of the Assembly of the Hebrews (Numbers 15:15) which is differing from being a sojourner, so that they could be entitled to Land inheritance just like Hebrews.
Some nations and classes of people were NOT allowed to ever convert to the Hebrews. Let's take a look:
YLT Dt 23:1 'One wounded, bruised, or cut in the member doth not enter into the assembly of Jehovah;
YLT Dt 23:2 a foreigner doth not enter into the assembly of Jehovah; even a tenth generation of him doth not enter into the assembly of Jehovah.
YLT Dt 23:3 'An Ammonite and a Moabite doth not enter into the assembly of Jehovah; even a tenth generation of them doth not enter into the assembly of Jehovah--to the age;
YLT Dt 23:4 because that they have not come before you with bread and with water in the way, in your coming out from Egypt, and because he hath hired against thee Balaam son of Beor, of Pethor of Aram-Naharaim, to revile thee;
YLT Dt 23:5 and Jehovah thy God hath not been willing to hearken unto Balaam, and Jehovah thy God doth turn for thee the reviling to a blessing, because Jehovah thy God hath loved thee;
YLT Dt 23:6 thou dost not seek their peace and their good all thy days--to the age.
As you can see from these verses, everything is pretty clear except maybe Deuteronomy 23:2. Let's take a look who the "foreigners" are...
YLT Lv 22:10 'And no stranger doth eat of the holy thing; a settler of a priest and an hireling doth not eat of the holy thing;
YLT Lv 22:11 and when a priest buyeth a person, the purchase of his money, he doth eat of it, also one born in his house; they do eat of his bread.
YLT Lv 22:12 'And a priest's daughter, when she is a strange man's,--she, of the heave-offering of the holy things doth not eat;
YLT Lv 22:13 and a priest's daughter, when she is a widow, or cast out, and hath no seed, and hath turned back unto the house of her father, as in her youth, of her father's bread she doth eat; but no stranger doth eat of it.
YLT Ex 12:43 And Jehovah saith unto Moses and Aaron, 'This is a statute of the passover; Any son of a stranger doth not eat of it;
YLT Ex 12:44 and any man's servant, the purchase of money, when thou hast circumcised him--then he doth eat of it;
YLT Ex 12:45 a settler or hired servant doth not eat of it;
YLT Lv 22:25 And from the hand of a son of a stranger ye do not bring near the bread of your God, of any of these, for their corruption is in them; blemish is in them; they are not pleasing for you.'
YLT Dt 14:21 'Ye do not eat of any carcase; to the sojourner who is within thy gates thou dost give it, and he hath eaten it; or sell it to a stranger; for a holy people thou art to Jehovah thy God; thou dost not boil a kid in its mother's milk.
YLT Dt 15:3 of the stranger thou mayest exact, and that which is thine with thy brother doth thy hand release;
So, as you can see, there are many types of people Torah mentions. And there are different rules for different groups, even though the requirement to the obedience of the Written Torah is similar to all groups.