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Sorry, I did not notice that your question was in $C^0$; so the following is a comment, not an answer. In the smooth ($C^\infty$) regularity class, or $C^r$ with $r\ge 1$, the restriction map is even a locally trivial fibration. You can replace $R^n$ by any manifold and $0$ by any submanifold. This was (a part of) Jean Cerf's first work in 1961.

Cerf, Jean Topologie de certains espaces de plongements.(French) Bull. Soc. Math. France 89 (1961), 227–380.

His "first fibration theorem" (p. 294, section 2.2.2, corollary 2) states that for $H\subset E\subset F$ (manifolds), the restriction map $Emb(E,F)\to Emb(H,F)$ is a locally trivial fibration. The theorem is actually not easy to find in the paper even if you read French perfectly, because of the formal style and of the extreme generality: the manifolds can have boundaries and corners, they can be noncompact, you can fix the embeddings on some subsets, he works with several differentiability classes, etc. Cerf was pretty young! In particular, if $E=F$ is noncompact, you can fix the germ at infinity of the embeddings $E\to E$ to be the identity; then, you get that the restriction map $Diff_c(E)\to Emb (H,E)$ is a locally trivial fibration, where $Diff_c(E)$ is the group of the compactly supported diffeomorphisms of $E$...

Sorry, I did not notice that your question was in $C^0$; so the following is a comment, not an answer. In the smooth ($C^\infty$) regularity class, or $C^r$ with $r\ge 1$, the restriction map is even a locally trivial fibre bundle (fibration). You can replace $R^n$ by any manifold and $0$ by any submanifold. This was (a part of) Jean Cerf's first work in 1961.

Cerf, Jean Topologie de certains espaces de plongements.(French) Bull. Soc. Math. France 89 (1961), 227–380.

His "first fibration theorem" (p. 294, section 2.2.2, corollary 2) states that for $H\subset E\subset F$ (manifolds), the restriction map $Emb(E,F)\to Emb(H,F)$ is a locally trivial fibre bundle. The theorem is actually not easy to find in the paper even if you read French perfectly, because of the formal style and of the extreme generality: the manifolds can have boundaries and corners, they can be noncompact, you can fix the embeddings on some subsets, he works with several differentiability classes, etc. Cerf was pretty young! In particular, if $E=F$ is noncompact, you can fix the germ at infinity of the embeddings $E\to E$ to be the identity; then, you get that the restriction map $Diff_c(E)\to Emb (H,E)$ is a locally trivial fibre bundle, where $Diff_c(E)$ is the group of the compactly supported diffeomorphisms of $E$...

Sorry, I did not notice that your question was in $C^0$; so the following is more a comment than an answer. In the smooth ($C^\infty$) regularity class, or $C^r$ with $r\ge 1$, the restriction map is even a locally trivial fibration. You can replace $R^n$ by any manifold and $0$ by any submanifold. This was (a part of) Jean Cerf's first work in 1961.

Cerf, Jean Topologie de certains espaces de plongements.(French) Bull. Soc. Math. France 89 (1961), 227–380.

His "first fibration theorem" (p. 294, section 2.2.2, corollary 2) states that for $H\subset E\subset F$ (manifolds), the restriction map $Emb(E,F)\to Emb(H,F)$ is a locally trivial fibration. The theorem is actually not easy to find in the paper even if you read French perfectly, because of the formal style and of the extreme generality: the manifolds can have boundaries and corners, they can be noncompact, you can fix the embeddings on some subsets, he works with several differentiability classes, etc. Cerf was pretty young! In particular, if $E=F$ is noncompact, you can fix the germ at infinity of the embeddings $E\to E$ to be the identity; then, you get that the restriction map $Diff_c(E)\to Emb (H,E)$ is a locally trivial fibration, where $Diff_c(E)$ is the group of the compactly supported diffeomorphisms of $E$...

Sorry, I did not notice that your question was in $C^0$; so the following is a comment, not an answer. In the smooth ($C^\infty$) regularity class, or $C^r$ with $r\ge 1$, the restriction map is even a locally trivial fibration. You can replace $R^n$ by any manifold and $0$ by any submanifold. This was (a part of) Jean Cerf's first work in 1961.

Cerf, Jean Topologie de certains espaces de plongements.(French) Bull. Soc. Math. France 89 (1961), 227–380.

His "first fibration theorem" (p. 294, section 2.2.2, corollary 2) states that for $H\subset E\subset F$ (manifolds), the restriction map $Emb(E,F)\to Emb(H,F)$ is a locally trivial fibration. The theorem is actually not easy to find in the paper even if you read French perfectly, because of the formal style and of the extreme generality: the manifolds can have boundaries and corners, they can be noncompact, you can fix the embeddings on some subsets, he works with several differentiability classes, etc. Cerf was pretty young! In particular, if $E=F$ is noncompact, you can fix the germ at infinity of the embeddings $E\to E$ to be the identity; then, you get that the restriction map $Diff_c(E)\to Emb (H,E)$ is a locally trivial fibration, where $Diff_c(E)$ is the group of the compactly supported diffeomorphisms of $E$...

It is even a locally trivial fibration. You can replace $R^n$ by any manifold and $0$ by any submanifold. This was Cerf's first work. Cerf, Jean Topologie de certains espaces de plongements.(French) Bull. Soc. Math. France 89 (1961), 227–380.

Sorry, I did not notice that your question was in $C^0$; so the following is more a comment than an answer. In the smooth ($C^\infty$) regularity class, or $C^r$ with $r\ge 1$, the restriction map is even a locally trivial fibration. You can replace $R^n$ by any manifold and $0$ by any submanifold. This was (a part of) Jean Cerf's first work in 1961.

Cerf, Jean Topologie de certains espaces de plongements.(French) Bull. Soc. Math. France 89 (1961), 227–380.

His "first fibration theorem" (p. 294, section 2.2.2, corollary 2) states that for $H\subset E\subset F$ (manifolds), the restriction map $Emb(E,F)\to Emb(H,F)$ is a locally trivial fibration. The theorem is actually not easy to find in the paper even if you read French perfectly, because of the formal style and of the extreme generality: the manifolds can have boundaries and corners, they can be noncompact, you can fix the embeddings on some subsets, he works with several differentiability classes, etc. Cerf was pretty young! In particular, if $E=F$ is noncompact, you can fix the germ at infinity of the embeddings $E\to E$ to be the identity; then, you get that the restriction map $Diff_c(E)\to Emb (H,E)$ is a locally trivial fibration, where $Diff_c(E)$ is the group of the compactly supported diffeomorphisms of $E$...